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Re: Integrate question
*To*: mathgroup at smc.vnet.net
*Subject*: [mg77649] Re: [mg77600] Integrate question
*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>
*Date*: Thu, 14 Jun 2007 05:32:01 -0400 (EDT)
*References*: <200706131144.HAA07285@smc.vnet.net>
On 13 Jun 2007, at 20:44, dimitris wrote:
> This question aose after replying in the forum for another CAS.
> In fact, it is not a question on "how I do something" but rather
> asking about some information about "why".
>
> In view of my reply I realized one fundamental difference
> between Integrate and the function Int from another CAS.
>
> int( diff(y(x),x) , x=a..b);
>
> b
> /
> | d
> | -- y(x) dx
> | dx
> /
> a
>
> In[11]:=
> Integrate[Derivative[1][f][x], {x, a, b}]
> Out[11]=
> -f[a]+f[b]
>
> I guess all we are familiar with the first fundamental theorem of
> calculus.
> It holds (http://mathworld.wolfram.com/
> FundamentalTheoremsofCalculus.html)
> if the function f[x] is continuous on the closed interval [a,b] (or at
> least piecewise
> continuous in Newton-Leibniz form with limits left and right of the
> finite discontinuities).
> I am with Mathematica default design. For example
>
> In[12]:=
> Integrate[Derivative[1][f][x]*Derivative[2][f][x], {x, a, b}]
>
> Out[12]=
> (1/2)*(-Derivative[1][f][a]^2 + Derivative[1][f][b]^2)
>
> In[13]:=
> Integrate[Derivative[1][f][x]*g[x] + f[x]*Derivative[1][g][x], {x, a,
> b}]
>
> Out[13]=
> (-f[a])*g[a] + f[b]*g[b]
>
> In[14]:=
> Integrate[Sin[f[x]]*Derivative[1][f][x], {x, a, b}]
>
> Out[14]=
> Cos[f[a]] - Cos[f[b]]
>
> I am neither software enginner, nor pure mathematician but this
> fundmental difference impressed me a lot! I am familiar
> with the issue of generic complex values in mathematica
> but here Mathematica "assumes" that typing Integrate[f'[x],{x,a,b}]
> f[x] is continuous in [a,b]?
>
> I will appreciate any comments on this issue.
>
> Greeting from the sunny Athens,
>
> Dimitris
>
>
Given a general functions f, without any special properties
Mathematica will assume that it is in fact an analytic function;
which is why you get:
Series[f[x], {x, 0, 3}]
SeriesData[x, 0, {f[0], Derivative[1][f][0], Derivative[2][f][0]/2,
Derivative[3][f][0]/6}, 0, 4, 1]
etc. So in particular, an undefined function is always infinitely
differentiable everywhere and hence continuous.
Series has an option, which can be used to remove this assumption:
Series[f[x]*Sin[x], {x, 0, 3}, Analytic -> False]
f[x]*SeriesData[x, 0, {1, 0, -1/6}, 1, 4, 1]
In other situations, e.g with Integrate, or D, there is no analogous
option but then I can't see any situation in which it could be useful
in the context of computer algebra. Can anyone?
Andrzej Kozlowski
although, to tell the truth, I can't think of any serious use for
this in the context of "computer algebra" (can anyone?)
Andrzej Kozlowski
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